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Full text of "History Of The Theory Of Numbers - I"

CHAP. V]
GENERALIZATIONS or ETJLER'S ^-FUNCTION.
149
these formulas for n = l, 2,..., n, we obtain two determinants of order n, each equal to (~l)n~lJk(n):
1-1-1        0-1        1      ..
1    _2*    -3*       0     -5*       6*    ..
0       1*       0     -2*       0     -3*    ..
00        1*     0        0   -2*   ..
lfc 2* 3* 4* ... 1111... 0101... 0010...
L. Gegenbauer204 proved (12).   For n=pi"1. . .p/8, set ir(n) = (-
where w(n) denotes the number of distinct prime factors of n.    By means of the series f (s) =Sn~8, he proved that, when d ranges over the divisors of r,
the last holding if r has no square factor and following from the third in view of (11),
=0 or
(mn2 == r) ,
according as r is or. is not a square,
where nly . . . , nt range over all sets of solutions of /c = l being due to H. G. Cantor.49
E. Cesaro169 derived (10) from (12), writing f1-fc for Jk.
E. Cesaro205 denoted Jk(ri) by \[/k(n) and gave (12).
L. Gegenbauer170 gave the further generalization
   nt+i = n, the case
J. Hammond206 wrote i/'(n, d) for S/(5), where / is an arbitrary function and 5 ranges over all multiples ^ n of the fixed divisor d of n.    Then
(13)                 2/() =^(n, 1) -2^(n, Pl) +S^(n, Plp2) - . . . ,
304Sitzungsber. Ak. Wiss. Wien (Math.), 89 II, 1884, 37-46.   Of. p. 841.   See Gegenbauer71
of Ch. X.
08Amiali di Mat., (2), 14, 1886-7, 142-6. ""Messenger Math., 20, 1890-1, 182-190.